Graded Picture Invariants and polynomial invariants for mixed tensor superspaces

Abstract

In this paper we consider the mixed tensor space of a $\mathbb Z_2$-graded vector space. We obtain a spanning set of invariants of the associated symmetric algebra under the action of the general linear supergroup as well as the queer supergroup over the Grassmann algebra. As a consequence, we give a generating set of polynomial invariants for the simultaneous adjoint action of the general linear supergroup on several copies of its Lie superalgebra. We show that in this special case, these turn out to be the supertrace monomials which is analogous to the results of Procesi in the classical case. A queer supergroup analogue of these results is also obtained.